Set Properties

A set is a collection of distinct objects or elements. Sets are often used in mathematics to represent groups of numbers, letters, or items that share a common property. Understanding the properties of sets helps in solving problems involving relationships, logic, and classification.

 

Basic Set Terminology

• Set: A group of elements, written in curly brackets. Example: $A = \{1, 2, 3\}$
• Element: An item in a set. Example: 2 is an element of $A$, written as $2 \in A$
• Empty set: A set with no elements. Written as $\emptyset$ or $\{\}$
• Universal set: The set that contains all elements under consideration. Often denoted by $U$
• Subset: Every element in one set is also in another. $A \subseteq B$

 

Set Notation

• $\in$: “is an element of”
• $\notin$: “is not an element of”
• $\subseteq$: “is a subset of”
• $\cup$: union (combine elements)
• $\cap$: intersection (common elements)
• $A'$ or $A^c$: complement of A (elements in the universal set but not in A)

 

Properties of Sets

1. Commutative Law

$$A \cup B = B \cup A \quad \text{and} \quad A \cap B = B \cap A$$

You can swap the sets — the order doesn’t change the result.

2. Associative Law

$$A \cup (B \cup C) = (A \cup B) \cup C \\ A \cap (B \cap C) = (A \cap B) \cap C$$

Grouping doesn’t change the result when taking union or intersection.

3. Distributive Law

$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$

Distributes like multiplication over addition in algebra!

 

Other Important Properties

• Idempotent Law: $A \cup A = A$, $A \cap A = A$
• Identity Law: $A \cup \emptyset = A$, $A \cap U = A$
• Complement Law: $A \cup A' = U$, $A \cap A' = \emptyset$
• Domination Law: $A \cup U = U$, $A \cap \emptyset = \emptyset$

 

Worked Example